Hyperbolic Geometry of Complex Network Data Konstantin Zuev http://www.its.caltech.edu/~zuev/ Joint work with D. Krioukov, M. Boguñá, and G. Bianconi CMX seminar, Caltech May 24, 2017
How do complex networks grow? Erdős Rényi Model G(n,p) 1. Take n nodes 2. Connect every pair of nodes at random with probability p Real networks are scale-free
Preferential Attachment Mechanism Barabási Albert Model 1. Start with n isolated nodes. New nodes come one at a time. 2. A new node i connects to m old nodes. 3. The probability that i connects to j i j Scale-free networks with rich gets richer Issues with PA Zero clustering No communities
Universal Properties of Complex Networks Heavy-tail degree distribution ( scale-free networks) Strong clustering ( many triangles ) Community structure PA Friendship network of children in a U.S. school
Popularity versus Similarity Intuition How does a new node make connections? It connects to popular nodes Preferential Attachment It connects to similar nodes Birds of feather flock together Homophily Key idea: new connections are formed by trade-off between popularity and similarity New node Popular node Similar node
Popularity-Similarity Model In a growing network: The popularity of node is modeled by its birth time The similarity of is modeled by distributed over a similarity space The angular distance quantifies the similarity between s and t. Mechanism: a new node connects to an existing node if is both popular and similar to, that is if: is small controls the relative contributions of popularity and similarity
Complex Network in a Hyperbolic Plane The angular coordinate of node is its similarity The radial coordinate of node is Let it grow with time: Let be the hyperbolic distance between and Minimization of minimization of New nodes connect to hyperbolically closets existing nodes Hyperbolic Disk We call this mechanism: Geometric Preferential Attachment
Geometric Preferential Attachment How does a new node find its position in the similarity space? Fashion: contains hot regions. Attractiveness of for a new node is the number of existing nodes in The higher the attractiveness of, the higher the probability that
GPA Model of Growing Networks 1. Initially the network is empty. New nodes appear one at a time. 2. The angular (similarity) coordinate of is determined as follows: a. Sample uniformly at random (candidate positions) b. Compute the attractiveness for all candidates c. Set with probability is initial attractiveness 3. The radial (popularity) coordinate of node is set to The radial coordinates of existing nodes are updated to models popularity fading 4. Node connects to hyperbolically closet existing nodes.
GPA as a Model for Real Networks GPA is the first model that generates networks with Power law degree distribution Strong clustering Community structure Universal properties of complex networks Can we estimate the model parameters from the network data? The model has three parameters: the number of links established by every new node the speed of popularity fading the initial attractiveness MLE
Hyper Map Embed into the hyperbolic plane Hyperbolic Atlas of the Internet Data: CAIDA Internet topology as of Dec 2009 Nodes are autonomous systems Two ASs are connected if they exchange traffic Node size Font size
References General text on Complex Networks M. Newman Networks: An Introduction 2009 aka Big Black Book Hyper Map F. Papadopoulos et al Network Mapping by Replaying Hyperbolic Growth IEEE/ACM Transactions on Networking, 2015 (first arxiv version 2012 ) Geometric Preferential Attachment K. Zuev et al Emergence of Soft Communities from Geometric Preferential Attachment Nature Scientific Reports 2015.